Maximum Directivity Factor Research Articles

A General Approach to the Design of the Fractional-Order Superdirective Beamformer

The superdirective beamformer (SD) with a linear array exhibits the maximum directivity factor (DF) in the isotropic noise field. Previously, a fractional-order SD (FO-SD) has been designed to achieve a compromise between the robustness and the DF. However, this method may suffer from two limitations. First, it only considers the Jacobi-Anger expansion-based beampattern synthesis. Second, it cannot control the DF in spherically isotropic noise field. To address these problems, we propose a general approach to the design of the FO-SD, which is a linear combination of two SDs with the integer order that can be designed with any beampattern synthesis method. Moreover, the proposed approach is able to control the WNG and the DF in both spherically and cylindrically isotropic noise field. Computer simulations substantiate the effectiveness of the proposed FO-SD over the existing SDs.

Maximum directivity of flanged open-ended waveguides

Directional beams have extensive applications in communication and sound reproduction. This paper investigates the theoretical maximum directivity of infinitely flanged open-ended waveguides and the radiation pattern synthesis. We derive a rigorous solution for the maximum directivity factor of a flanged aperture with arbitrary shape by projecting its surface velocity on the waveguide modes, enabling the creation of a directional beam in any desired direction. We present case studies for a three-dimensional circular waveguide and a bidimensional waveguide. The theoretical beam that is obtained in a subspace spanned by all the propagating modes can then be synthesized by a group of incident modes or a point-source array within the waveguide. The optimality of the beam is demonstrated by comparing it with Gaussian shaded modes radiated from the waveguide. If the evanescent modes are taken into account, the maximum directivity factor increases with considerable loss to the radiation efficiency. Nevertheless, the optimum aperture velocity dominated by its evanescent components is capable of precise beam steering in extreme directions and could be useful for designing material-filled horns. Our work provides benchmark directivity factors and patterns for the practical design of horn antennas. In addition, we present a generalized form of Bouwkamp's impedance theorem.

Design of Maximum Directivity Beamformers With Linear Acoustic Vector Sensor Arrays

This paper studies the design of maximum directivity factor (MDF) beamformers based on uniform linear arrays (ULAs) consisting of acoustic vector sensors (AVSs). We first derive the main lobe constraints, which ensure that the beamformer's beampattern achieves a maximum in the look direction, and prove that any beamformer that satisfies the proposed constraints can be written as the sum of two orthogonal beamformers: the maximum white noise gain (MWNG) beamformer and a reduced-rank beamformer. Then, we derive the MDF beamformer by maximizing the directivity factor (DF) under the deduced constraints. We also derive a robust version of the MDF beamformer, which can keep the WNG above a pre-specified level. Compared to the conventional MDF beamformer based on ULAs with omnidirectional microphones, the designed MDF beamformer with uniform linear AVS arrays (ULAVSAs) can steer the beampattern to any look direction in the 3-dimensional space and achieves a higher directivity. The proposed MDF beamformer also outperforms the two-step MDF beamformer with ULAVSAs since it maximizes the DF. The proposed methods are validated through simulations as well as real experiments.

Beamforming with Cube Microphone Arrays Via Kronecker Product Decompositions

Microphone arrays combined with beamforming have been widely used to solve many important acoustic problems in a wide range of applications. Much effort has been devoted in the literature to microphone array beamforming, among which the Kronecker product beamforming method developed recently has demonstrated some interesting properties. Generally, this method decomposes the global beamforming filter into a Kronecker product of a number of sub-beamforming filters, each of which corresponds to a virtual subarray and can be designed individually. This decomposition not only reduces significantly the number of beamforming coefficients, but also can be explored to improve the robustness and flexibility of beamforming. This paper extends Kronecker product beamforming from two-dimensional arrays into three-dimensional cube arrays. We consider two decompositions, i.e., fully and partially separable ones. The former decomposes the entire array into three linear subarrays while the latter decomposes the entire array into a linear subarray and a planar one. Then, for each case, we derive the Kronecker product maximum white noise gain beamformer, the Kronecker product approximate maximum directivity factor (DF) beamformer, the Kronecker product null-steering beamformer, and the Kronecker product iterative maximum DF beamformer. Simulation results demonstrate the properties and advantages of the proposed beamformers.

Superdirectivity solutions of circular arrays with acoustic particle velocity sensors

This paper presents analytical superdirectivity solutions for circular arrays with acoustic particle velocity sensors (APVSs) based on the eigenbeam decomposition and synthesis theory. The main findings are summarized as follows. (1) The optimal beampattern, the global directivity factor, and the total error sensitivity function of circular APVS arrays, like those of circular arrays with pressure sensors, can also be expressed as the sums of eigenbeams, their directivity factors, and their error sensitivity functions, respectively. (2) The directivity factor increases but robustness decreases with an increase in the order number at low frequencies, and the reduced-rank technique can be used to synthesize robust superdirective beampatterns by truncating some high-order eigenbeams. (3) The limit theoretical maximum directivity factor and optimal beampattern are derived, which provide the upper bound of performance for circular APVS arrays and show that this type of arrays also has good potential for improving directivity. The reduced–rank technique is applied for circular APVS arrays to obtain robust high-order superdirectivity in simulations, which shows good performance for improving the robustness.

A Simple Theory and New Method of Differential Beamforming With Uniform Linear Microphone Arrays

This article presents a theoretical study of differential beamforming with uniform linear arrays. By defining a forward spatial difference operator, any order of the spatial difference of the observed signals can be represented as a product of a difference operator matrix and the microphone array observations. Consequently, differential beamforming is implemented in two stages, where the first one obtains spatial difference of the observations and the second stage optimizes the beamformer. The major contributions of this article are as follows. First, we propose a new theory of differential beamforming with uniform linear arrays, which shows clearly the connection between the conventional differential beamforming and the null-constrained differential beamforming methods. This provides some new insight into the design of differential beamformers. Second, we deduce some new differential beamformers, where conventional beamforming may be seen as a particular case. Specifically, we derive the maximum white noise gain (MWNG), maximum directivity factor (MDF), parameterized MDF, and parameterized maximum front-to-back ratio differential beamformers. Third, we further extend the idea of how to design optimal differential beamformers by combining both the observed signals and their spatial differences.

Differential Beamforming on Graphs

In this article, we study differential beamforming from a graph perspective. The microphone array used for differential beamforming is viewed as a graph, where its sensors correspond to the nodes, the number of microphones corresponds to the order of the graph, and linear spatial difference equations among microphones are related to graph edges. Specifically, for the first-order differential beamforming with an array of $M$ microphones, each pair of adjacent microphones are directly connected, resulting in $M-1$ spatial difference equations. On a graph, each of these equations corresponds to a 2-clique. For the second-order differential beamforming, each three adjacent microphones are directly connected, resulting in $M-2$ second-order spatial difference equations, and each of these equations corresponds to a 3-clique. In an analogous manner, the differential microphone array for any order of differential beamforming can be viewed as a graph. From this perspective, we then derive a class of differential beamformers, including the maximum white noise gain beamformer, the maximum directivity factor one, and optimal compromising beamformers. Simulations are presented to demonstrate the performance of the derived differential beamformers.

Constrained minimum power combination for broadband beamformer design in the STFT domain

Beamforming using sensor array is widely used in spatial signal processing since it offers better spatial focusing capability than single sensor. However, in practical applications for broadband signal, there always exists a trade-off issue between the directivity capability of an array and its robustness on system errors. In this paper, in order to combine merits of different beamformers instead of trade-off their performances, we propose a constrained minimum-power combination method. We firstly analyze two optimal beamformers that maximize Directivity Factor (DF) and White Noise Gain (WNG) respectively. Then we propose a non-linear combination method, which automatically selects the best beamformer that has the minimum output power, so as to control the unwanted white noise amplification and keep the maximum DF if possible. Two solutions to the proposed combination strategy are given. They do not need to determine the correct trade-off factor used in linear combination method, and avoid challenge estimations on noise and target statistics required in adaptive beamforming. The performance of the proposed beamformer is evaluated in ideal noise fields and complicated noise fields respectively. It is shown that the proposed beamformer integrates merits of different beamformers. It always achieves the best speech quality and biggest noise reduction compared to other popular beamformers.

Superdirective Beamforming Based on the Krylov Matrix

Superdirective beamforming has attracted a significant amount of research interest in speech and audio applications, since it can maximize the directivity factor (DF) given an array geometry and, therefore, is efficient in dealing with signal acquisition in diffuse-like noise environments. However, this beamformer is very sensitive to sensor self-noise and mismatch among sensors, which considerably restricts its use in practical systems. This paper develops an approach to superdirective beamforming based on the Krylov matrix. We show that the columns of a proposed Krylov matrix, which span a chosen dimension of the whole space, are interesting beamformers; consequently, all different linear combinations of those columns lead to beamformers that have good properties. In particular, we develop the Krylov maximum white noise gain and Krylov maximum DF beamformers, which are obtained by maximizing the WNG and the DF, respectively. By properly choosing the dimension of the Krylov subspace, the developed beamformers that can make a compromise between reasonable values of the DF and white noise amplification. We also extend the basic idea to the design of the Krylov maximum front-to-back ratio, parametric superdirective, and parametric supercardioid beamformers.

A general superdirectivity model for arbitrary sensor arrays

This paper proposes a general model of superdirectivity to provide analytical and closed-form solutions for arbitrary sensor arrays. Based on the equivalence between the maximum directivity factor and the maximum array gain in the isotropic noise field, Gram-Schmidt orthogonalization is introduced and recursively transformed into a matrix form to conduct pre-whitening and matching operations that result in superdirectivity solutions. A Gram-Schmidt mode-beam decomposition and synthesis method is then presented to formally implement these solutions. Illustrative examples for different arrays are provided to demonstrate the feasibility of this method, and a reduced rank technique is used to deal with the practical array design for robust beamforming and acceptable high-order superdirectivity. Experimental results that are provided for a linear array consisting of nine hydrophones show the good performance of the technique. A superdirective beampattern with a beamwidth of 48.05° in the endfire direction is typically achieved when the inter-sensor spacing is only 0.09λ (λ is the wavelength), and the directivity index is up to 12 dB, which outperforms that of the conventional delay-and-sum counterpart by 6 dB.

Robust high-order superdirectivity of circular sensor arrays.

This paper presents a detailed study of the high-order superdirectivity of circular sensor arrays, which is aimed at completing the authors' recently proposed analytical superdirectivity model. From the limit expression of the maximum directivity factor, it is shown that the circular arrays possess good potential for directivity improvement. It is found that the sensitivity function used as a robustness measurement can also be accurately decomposed into a series of closed-form sensitivity functions of eigenbeams, similar to the optimal beampattern and its corresponding directivity factor. Moreover, the performance of eigenbeams can be regarded as an indicator of error sensitivity, and the robustness constraint parameters can be estimated easily. Two specific approaches are proposed for obtaining robust superdirectivity on the basis of robustness analyses, and their performance is demonstrated experimentally.

Directivity factors for linear arrays of velocity sensors

Some of the features unique to beamforming a linear array of acoustic velocity sensors, which are not present with scalar-sensing elements (such as conventional pressure sensors), are described in this paper. Four types of sensors are considered here: a uniaxial motion sensor, which measures acoustic particle velocity along a single axis; a biaxial motion sensor measuring velocity in two orthogonal directions; a triaxial motion sensor that measures all three orthogonal components of the velocity vector; and a sensor, denoted as an acoustic vector sensor, that measures acoustic pressure as well as the complete velocity vector. Comparisons are made of the directivity index for each type of sensor and for linear arrays of sensors. It is shown that uniaxial velocity sensors can have a maximum directivity factor three times greater than an omnidirectional pressure sensor, a gain in directivity index of 4.8 dB. Not surprisingly, this directivity gain is highly dependent on signal arrival direction. Indeed, a uniaxial velocity sensor’s directivity can be less than that of an omnidirectional pressure, indicative of a loss in signal level. These comparisons further indicate that a single vector sensor can provide 6 dB of directivity gain, four times the directivity of a pressure sensor. Line arrays of directional sensors can have a directivity index approximately 5 dB greater than that of an identical line array of pressure sensors for approximately all azimuthal array steerings.

A new method of directivity synthesis

A completely new synthesis method, called the successive approximation method, is developed in this article. By introducing a control function and using a step-by-step approach program, various directivity functions can be obtained in which the envelope of the peaks of the sidelobes fits with a specified curve, with equisidelobe directivity functions, and with the directivity functions corresponding to the maximum directivity factor for specified sidelobe levels. This method is applicable for various forms of arrays and can be used with amplitude shading, phase shading, and unequally spaced arrays. Numerical calculations have been made for linear arrays to permit study of the directivity functions with the maximum directivity factor for specified sidelobe levels. Then, these directivity functions are compared with equisidelobe directivity functions. Calculations show that, when specified sidelobe level is low enough, the directivity function with the maximum directivity factor is just the equisidelobe directivity function.

Constrained optimization of monopulse circular aperture distribution in the presence of blockage

A method of synthesizing aperture antenna excitations to obtain optimum performance in the presence of aperture blockage is presented. The problem of optimizing the performance indices such as directivity and angular sensitivity is formulated as an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -dimensional minimization problem, with constraint on the sidelobe level. In most cases, the objective function is nonlinear with multiple minima which does not yield readily to gradient methods. A simplex search method is used in the study for optimizing the performance index. The sidelobe level (SLL) constraint has been incorporated using the penalty function method. The method is applied to circular monopulse aperture distribution with circular blockage to obtain maximum directivity factor (DF) in the sum mode and maximum angular sensitivity factor (ASF) in the difference mode, with sidelobe level constraint. The effect of blockage on the maximum directivity and maximum angular sensitivity is studied for various sidelobe level constraints.

Optimum aperture distributions in the presence of blockage

A method is presented for optimising the performance indices of aperture antennas in the presence of blockage. An N-dimensional objective function is formed for maximising the directivity factor of a circular aperture with blockage under sidelobe-level constraints, and is minimised using the simplex search method. Optimum aperture distributions are computed for a circular aperture with blockage of circular geometry that gives the maximum directivity factor under sidelobe-level constraints.

Maximum directivity factor of nth-order transducers

It is proved using general group-theoretic and Hilbert-space methods that the maximum directivity factor obtainable from an nth-order directional acoustic transducer is (n+1)2, that this most directional transducer is axially symmetric, and that its polar response to sounds arriving from an angle ϑ off axis is proportional to 𝒥n/k=0 (2k+1) Pk(cosϑ). Subject Classification: [43]85.40, [43]85.62.

Second-order gradient unidirectional microphones utilizing an electret transducer

A class of second-order gradient microphones with unidirectional directivity characteristics is analyzed. These microphones consist of four in-line sensors whose outputs are added with proper delays. The directivity characteristics of such microphones are rotationally symmetric and given by (α+cosϑ)(β+cosϑ), where α and β are constants and ϑ is the angle of incidence relative to the rotational axis. The calculated maximum directivity factor is 9. Experimental systems consist of an electret transducer enclosed in a cylindrical cavity and four tubes, paired in length, extending from the cavity to sample the sound field. Measurements on a transducer with α=1 and ‖β‖ ≪ 1 yield a sensitivity of about −80 dBV/μbar in the direction of mzximum pickup (ϑ=0) with variations of less than ±3 dB in the frequency range from 0.2 to 4 kHz. The measured sensitivity is about 20 dB lower for ϑ=90, 180 and 270°. Because of their large directivity factors, such microphones are of interest in environments with high noise levels or in situations where reverberation or acoustic feedback have to be suppressed. Subject Classification: 85.62, 85.44, 85.40.